Question 1

Evaluate the integral by using a substitution prior to integration by parts.
-$$\int _ { 0 } ^ { 1 } \frac { x } { \sqrt { x + 1 } } d x$$

Accepted Answer

A)  $- 1.33$ 
B)  $- 2.27$ 
C)  $- 0.94$ 
D)  $0.39$ 
A)  $- 1.33$ 
B)  $- 2.27$ 
C)  $- 0.94$ 
D)  $0.39$

Question 2

Evaluate the integral.
-$\int 2 x e ^ { x } d x$

Accepted Answer

A)  $x e ^ { x } - 2 e ^ { x } + C$ 
B)  $2 e ^ { x } - 2 x e ^ { x } + C$ 
C)  $2 x e ^ { x } - 2 e ^ { x } + C$ 
D)  $2 e ^ { x } - e ^ { x } + C$ 
A)  $x e ^ { x } - 2 e ^ { x } + C$ 
B)  $2 e ^ { x } - 2 x e ^ { x } + C$ 
C)  $2 x e ^ { x } - 2 e ^ { x } + C$ 
D)  $2 e ^ { x } - e ^ { x } + C$

Question 3

Find the volume of the solid generated by revolving the region bounded by the curve $y = \ln x$, the $x$-axis, and the vertical line $\mathrm { x } = \mathrm { e } ^ { 2 }$ about the $\mathrm { x }$-axis.

Accepted Answer

A)  $\pi ( e - 1 )$ 
B)  $\pi \mathrm { e }$ 
C)  $\pi \left( e ^ { 2 } - 1 \right)$ 
D)  $2 \pi \left( e ^ { 2 } - 1 \right)$ 
A)  $\pi ( e - 1 )$ 
B)  $\pi \mathrm { e }$ 
C)  $\pi \left( e ^ { 2 } - 1 \right)$ 
D)  $2 \pi \left( e ^ { 2 } - 1 \right)$

Question 4

Evaluate the integral.
-$\int _ { 0 } ^ { 1 / 4 } y \tan ^ { - 1 } 4 y \mathrm { dy } \quad$ (Give your answer in exact form.)

Accepted Answer

A)  $\frac { \pi } { 4 } - \frac { 1 } { 2 }$ 
B)  $\frac { \pi } { 64 } - \frac { 1 } { 32 }$ 
C)  $\frac { \pi } { 128 } - \frac { 1 } { 32 }$ 
D)  $\frac { 1 } { 64 }$ 
A)  $\frac { \pi } { 4 } - \frac { 1 } { 2 }$ 
B)  $\frac { \pi } { 64 } - \frac { 1 } { 32 }$ 
C)  $\frac { \pi } { 128 } - \frac { 1 } { 32 }$ 
D)  $\frac { 1 } { 64 }$

Question 5

Integrate the function.
-$$\int \frac { \left( 49 - t ^ { 2 } \right) ^ { 3 / 2 } } { t ^ { 6 } } d t$$

Accepted Answer

A)  $- \frac { 1 } { 245 } \left( \frac { \sqrt { 49 - t ^ { 2 } } } { t } \right) ^ { 5 } + \sec ^ { - 1 } \left( \frac { t } { 7 } \right) + C$ 
B)  $- \frac { \left( 49 - t ^ { 2 } \right) ^ { 5 / 2 } } { 245 t ^ { 7 } }$ 
C)  $- \frac { 1 } { 5 } \left( \frac { \sqrt { 49 - t ^ { 2 } } } { t } \right) ^ { 5 } + C$ 
D)  $- \frac { 1 } { 245 } \left( \frac { \sqrt { 49 - t ^ { 2 } } } { t } \right) ^ { 5 } + C$ 
A)  $- \frac { 1 } { 245 } \left( \frac { \sqrt { 49 - t ^ { 2 } } } { t } \right) ^ { 5 } + \sec ^ { - 1 } \left( \frac { t } { 7 } \right) + C$ 
B)  $- \frac { \left( 49 - t ^ { 2 } \right) ^ { 5 / 2 } } { 245 t ^ { 7 } }$ 
C)  $- \frac { 1 } { 5 } \left( \frac { \sqrt { 49 - t ^ { 2 } } } { t } \right) ^ { 5 } + C$ 
D)  $- \frac { 1 } { 245 } \left( \frac { \sqrt { 49 - t ^ { 2 } } } { t } \right) ^ { 5 } + C$

Question 6

Evaluate the integral.
-$$\int \cos ^ { - 1 } x d x$$

Accepted Answer

A)  $x \cos ^ { - 1 } x - \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } + C$ 
B)  $x \cos ^ { - 1 } x - \sqrt { 1 - x ^ { 2 } } + C$ 
C)  $x \cos ^ { - 1 } x - 2 \sqrt { 1 - x ^ { 2 } } + C$ 
D)  $x \cos ^ { - 1 } x + \sqrt { 1 - x ^ { 2 } } + C$ 
A)  $x \cos ^ { - 1 } x - \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } + C$ 
B)  $x \cos ^ { - 1 } x - \sqrt { 1 - x ^ { 2 } } + C$ 
C)  $x \cos ^ { - 1 } x - 2 \sqrt { 1 - x ^ { 2 } } + C$ 
D)  $x \cos ^ { - 1 } x + \sqrt { 1 - x ^ { 2 } } + C$

Question 7

Integrate the function.
-$$\int \frac { d x } { \left( 25 x ^ { 2 } + 1 \right) ^ { 2 } }$$

Accepted Answer

A)  $\frac { 5 x } { 25 x ^ { 2 } + 1 } + C$ 
B)  $\frac { 1 } { 10 } \tan ^ { - 1 } 5 x + \frac { x } { 50 x ^ { 2 } + 2 } + C$ 
C)  $\ln \left| \sqrt { 25 x ^ { 2 } + 1 } + 5 x \right| + C$ 
D)  $\tan ^ { - 1 } 5 x - \frac { 5 x } { 25 x ^ { 2 } + 1 } + C$ 
A)  $\frac { 5 x } { 25 x ^ { 2 } + 1 } + C$ 
B)  $\frac { 1 } { 10 } \tan ^ { - 1 } 5 x + \frac { x } { 50 x ^ { 2 } + 2 } + C$ 
C)  $\ln \left| \sqrt { 25 x ^ { 2 } + 1 } + 5 x \right| + C$ 
D)  $\tan ^ { - 1 } 5 x - \frac { 5 x } { 25 x ^ { 2 } + 1 } + C$

Question 8

Use integration by parts to establish a reduction formula for the integral.
-$\int ( \ln a x ) ^ { n } d x$

Accepted Answer

A)  $\int ( \ln a x ) ^ { \mathrm { n } } \mathrm { dx } = a x ( \ln a x ) ^ { \mathrm { n } } - a n \int ( \ln a x ) ^ { \mathrm { n } - 1 } \mathrm { dx }$ 
B)  $\int ( \ln a x ) ^ { n } d x = x ( \ln a x ) ^ { n } - n \int ( \ln a x ) ^ { n - 1 } d x$ 
C)  $\int ( \ln a x ) ^ { n } d x = \frac { x ( \ln a x ) ^ { n } } { n } + \frac { n } { a } \int ( \ln a x ) ^ { n - 1 } d x$ 
D)  $\int ( \ln a x ) ^ { n } d x = \frac { x ( \ln a x ) ^ { n } } { n } - \frac { n } { a } \int ( \ln a x ) ^ { n - 2 } d x$ 
A)  $\int ( \ln a x ) ^ { \mathrm { n } } \mathrm { dx } = a x ( \ln a x ) ^ { \mathrm { n } } - a n \int ( \ln a x ) ^ { \mathrm { n } - 1 } \mathrm { dx }$ 
B)  $\int ( \ln a x ) ^ { n } d x = x ( \ln a x ) ^ { n } - n \int ( \ln a x ) ^ { n - 1 } d x$ 
C)  $\int ( \ln a x ) ^ { n } d x = \frac { x ( \ln a x ) ^ { n } } { n } + \frac { n } { a } \int ( \ln a x ) ^ { n - 1 } d x$ 
D)  $\int ( \ln a x ) ^ { n } d x = \frac { x ( \ln a x ) ^ { n } } { n } - \frac { n } { a } \int ( \ln a x ) ^ { n - 2 } d x$

Question 9

Solve the problem.
-Find the length of the curve $y = \ln ( \sin x ) , \pi / 3 \leq x \leq \pi / 2$

Accepted Answer

A)  $\ln ( \sqrt { 3 } + 1 )$ 
B)  $1 - \ln ( \sqrt { 3 } )$ 
C)  $\ln ( \sqrt { 3 } )$ 
D)  $\ln ( 2 \sqrt { 3 } )$ 
A)  $\ln ( \sqrt { 3 } + 1 )$ 
B)  $1 - \ln ( \sqrt { 3 } )$ 
C)  $\ln ( \sqrt { 3 } )$ 
D)  $\ln ( 2 \sqrt { 3 } )$

Question 10

Evaluate the integral.
-$$\int _ { 0 } ^ { \pi / 2 } \cos ^ { 2 } 3 x \sin ^ { 3 } 3 x d x$$

Accepted Answer

A)  $\frac { 2 } { 45 }$ 
B)  0 
C)  $\frac { 1 } { 45 }$ 
D)  $\frac { 8 } { 45 }$ 
A)  $\frac { 2 } { 45 }$ 
B)  0 
C)  $\frac { 1 } { 45 }$ 
D)  $\frac { 8 } { 45 }$

Question 11

Evaluate the integral.
-$$\int _ { 0 } ^ { \pi / 2 } \sin 4 t \sin 3 t d t$$

Accepted Answer

A)  $\frac { 8 } { 15 }$ 
B)  $\frac { 5 } { 7 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { 4 } { 7 }$ 
A)  $\frac { 8 } { 15 }$ 
B)  $\frac { 5 } { 7 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { 4 } { 7 }$

Question 12

Integrate the function.
-$$\int \frac { d x } { x \sqrt { 36 x ^ { 2 } - 49 } }$$

Accepted Answer

A)  $\frac { 1 } { 7 } \sec ^ { - 1 } \frac { 6 } { 7 } x + C$ 
B)  $\frac { 6 } { 7 } \sec ^ { - 1 } \frac { 6 } { 7 } x + C$ 
C)  $\frac { 6 } { 7 } \sin ^ { - 1 } \frac { 6 } { 7 } x + C$ 
D)  $\frac { 1 } { 6 } \sin ^ { - 1 } \frac { 6 } { 7 } x + C$ 
A)  $\frac { 1 } { 7 } \sec ^ { - 1 } \frac { 6 } { 7 } x + C$ 
B)  $\frac { 6 } { 7 } \sec ^ { - 1 } \frac { 6 } { 7 } x + C$ 
C)  $\frac { 6 } { 7 } \sin ^ { - 1 } \frac { 6 } { 7 } x + C$ 
D)  $\frac { 1 } { 6 } \sin ^ { - 1 } \frac { 6 } { 7 } x + C$

Question 13

Evaluate the integral.
-Use the formula $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int f ( y ) d y , y = f ^ { - 1 } ( x )$ to evaluate the integral. $\int \cos ^ { - 1 } x d x$

Accepted Answer

A)  $x \cos ^ { - 1 } x - \sin \left( \cos ^ { - 1 } x \right) + C$ 
B)  $x \cos ^ { - 1 } x - \sin x + C$ 
C)  $x \cos ^ { - 1 } x + \sin \left( \cos ^ { - 1 } x \right) + C$ 
D)  $x \cos ^ { - 1 } x - x + C$ 
A)  $x \cos ^ { - 1 } x - \sin \left( \cos ^ { - 1 } x \right) + C$ 
B)  $x \cos ^ { - 1 } x - \sin x + C$ 
C)  $x \cos ^ { - 1 } x + \sin \left( \cos ^ { - 1 } x \right) + C$ 
D)  $x \cos ^ { - 1 } x - x + C$

Question 14

Use integration by parts to establish a reduction formula for the integral.
-$$\int \sec ^ { n } x d x , n \neq 1$$

Accepted Answer

A)  $\int \sec ^ { n } x d x = \sec ^ { n - 2 } x \tan x - ( n - 2 ) \int \sec ^ { n - 2 } x \tan x d x$ 
B)  $\int \sec ^ { n } x d x = \frac { 1 } { n - 1 } \sec ^ { n - 2 } x \tan x + \frac { n - 2 } { n - 1 } \int \sec ^ { n - 2 } x d x$ 
C)  $\int \sec ^ { n } x d x = \sec ^ { n - 2 } x \tan x + ( n - 2 ) \int \sec ^ { n - 2 } x d x$ 
D)  $\int \sec ^ { n } x d x = \frac { 1 } { n } \sec ^ { n } x \tan x - \frac { n - 1 } { n } \int \sec ^ { n - 1 } x d x$ 
A)  $\int \sec ^ { n } x d x = \sec ^ { n - 2 } x \tan x - ( n - 2 ) \int \sec ^ { n - 2 } x \tan x d x$ 
B)  $\int \sec ^ { n } x d x = \frac { 1 } { n - 1 } \sec ^ { n - 2 } x \tan x + \frac { n - 2 } { n - 1 } \int \sec ^ { n - 2 } x d x$ 
C)  $\int \sec ^ { n } x d x = \sec ^ { n - 2 } x \tan x + ( n - 2 ) \int \sec ^ { n - 2 } x d x$ 
D)  $\int \sec ^ { n } x d x = \frac { 1 } { n } \sec ^ { n } x \tan x - \frac { n - 1 } { n } \int \sec ^ { n - 1 } x d x$

Question 15

Use various trigonometric identities to simplify the expression then integrate.
-$$\int \sin \theta \cos \theta \cos 5 \theta d \theta$$

Accepted Answer

A)  $\frac { 1 } { 28 } \cos 4 \theta + C$ 
B)  $\frac { 1 } { 12 } \cos 3 \theta - \frac { 1 } { 28 } \cos 7 \theta + C$ 
C)  $\frac { 1 } { 7 } \cos 7 \theta + C$ 
D)  $\frac { 1 } { 5 } \cos 3 \theta - \frac { 1 } { 10 } \cos 7 \theta + C$ 
A)  $\frac { 1 } { 28 } \cos 4 \theta + C$ 
B)  $\frac { 1 } { 12 } \cos 3 \theta - \frac { 1 } { 28 } \cos 7 \theta + C$ 
C)  $\frac { 1 } { 7 } \cos 7 \theta + C$ 
D)  $\frac { 1 } { 5 } \cos 3 \theta - \frac { 1 } { 10 } \cos 7 \theta + C$

Question 16

Find the area between $y = ( x - 3 ) e ^ { x }$ and the $x$-axis from $x = 3$ to $x = 6$.

Accepted Answer

A)  $\mathrm { e } ^ { 6 } - \mathrm { e } ^ { 3 }$ 
B)  $\mathrm { e } ^ { 6 } + \mathrm { e } ^ { 3 }$ 
C)  $2 \mathrm { e } ^ { 6 }$ 
D)  $2 \mathrm { e } ^ { 6 } + \mathrm { e } ^ { 3 }$ 
A)  $\mathrm { e } ^ { 6 } - \mathrm { e } ^ { 3 }$ 
B)  $\mathrm { e } ^ { 6 } + \mathrm { e } ^ { 3 }$ 
C)  $2 \mathrm { e } ^ { 6 }$ 
D)  $2 \mathrm { e } ^ { 6 } + \mathrm { e } ^ { 3 }$

Question 17

Evaluate the integral.
-$\int x ^ { 3 } \ln 4 x d x$

Accepted Answer

A)  $\frac { 1 } { 4 } x ^ { 4 } \ln 4 x - \frac { 1 } { 20 } x ^ { 5 } + C$ 
B)  $\frac { 1 } { 4 } x ^ { 4 } \ln 4 x - \frac { 1 } { 16 } x ^ { 4 } + C$ 
C)  $\frac { 1 } { 4 } x ^ { 4 } \ln 4 x + \frac { 1 } { 16 } x ^ { 4 } + C$ 
D)  $\ln 4 x - \frac { 1 } { 4 } x ^ { 4 } + C$ 
A)  $\frac { 1 } { 4 } x ^ { 4 } \ln 4 x - \frac { 1 } { 20 } x ^ { 5 } + C$ 
B)  $\frac { 1 } { 4 } x ^ { 4 } \ln 4 x - \frac { 1 } { 16 } x ^ { 4 } + C$ 
C)  $\frac { 1 } { 4 } x ^ { 4 } \ln 4 x + \frac { 1 } { 16 } x ^ { 4 } + C$ 
D)  $\ln 4 x - \frac { 1 } { 4 } x ^ { 4 } + C$

Question 18

Evaluate the integral.
-$\int y ^ { 3 } e ^ { - 3 y } d y$

Accepted Answer

A)  $- \mathrm { e } ^ { - 3 \mathrm { y } } \left[ \frac { 1 } { 3 } \mathrm { y } ^ { 3 } + \frac { 1 } { 3 } \mathrm { y } ^ { 2 } + \frac { 2 } { 9 } \mathrm { y } + \frac { 2 } { 27 } \right] + \mathrm { C }$ 
B)  $\mathrm { e } ^ { - 3 \mathrm { y } } \left[ \frac { 1 } { 3 } \mathrm { y } ^ { 3 } - \frac { 1 } { 3 } \mathrm { y } ^ { 2 } + \frac { 2 } { 9 } \mathrm { y } - \frac { 2 } { 27 } \right] + \mathrm { C }$ 
C)  $- \frac { 1 } { 12 } y ^ { 4 } e ^ { - 3 y } + C$ 
D)  $- \frac { 1 } { 3 } e ^ { - 3 y } \left[ y ^ { 3 } + y ^ { 2 } + y + 6 \right] + C$ 
A)  $- \mathrm { e } ^ { - 3 \mathrm { y } } \left[ \frac { 1 } { 3 } \mathrm { y } ^ { 3 } + \frac { 1 } { 3 } \mathrm { y } ^ { 2 } + \frac { 2 } { 9 } \mathrm { y } + \frac { 2 } { 27 } \right] + \mathrm { C }$ 
B)  $\mathrm { e } ^ { - 3 \mathrm { y } } \left[ \frac { 1 } { 3 } \mathrm { y } ^ { 3 } - \frac { 1 } { 3 } \mathrm { y } ^ { 2 } + \frac { 2 } { 9 } \mathrm { y } - \frac { 2 } { 27 } \right] + \mathrm { C }$ 
C)  $- \frac { 1 } { 12 } y ^ { 4 } e ^ { - 3 y } + C$ 
D)  $- \frac { 1 } { 3 } e ^ { - 3 y } \left[ y ^ { 3 } + y ^ { 2 } + y + 6 \right] + C$

Question 19

Find the volume of the solid generated by revolving the region bounded by the curve $y = 2 \cos x$ and the $x$-axis, $\frac { \pi } { 2 } \leq x \leq \frac { 3 \pi } { 2 }$, about the $x$-axis.

Accepted Answer

A)  $2 \pi ^ { 2 }$ 
B)  $4 \pi ^ { 2 }$ 
C)  $2 \pi ^ { 3 }$ 
D)  $4 \pi$ 
A)  $2 \pi ^ { 2 }$ 
B)  $4 \pi ^ { 2 }$ 
C)  $2 \pi ^ { 3 }$ 
D)  $4 \pi$

Question 20

Find the volume of the solid generated by revolving the region in the first quadrant bounded by $y = e ^ { x }$ and the $x$-axis, from $x = 0$ to $x = \ln 3$, about the $y$-axis.

Accepted Answer

A)  $3 \pi \ln 3$ 
B)  $6 \pi \ln 3$ 
C)  $2 \pi ( 3 \ln 3 - 3 )$ 
D)  $2 \pi ( 3 \ln 3 - 2 )$ 
A)  $3 \pi \ln 3$ 
B)  $6 \pi \ln 3$ 
C)  $2 \pi ( 3 \ln 3 - 3 )$ 
D)  $2 \pi ( 3 \ln 3 - 2 )$

Evaluate the integral by using a substitution prior to integration by parts. - $\int _ { 0 } ^ { 1 } \frac { x } { \sqrt { x + 1 } } d x$

Evaluate the integral. - $\int 2 x e ^ { x } d x$

Find the volume of the solid generated by revolving the region bounded by the curve $y = \ln x$ , the $x$ -axis, and the vertical line $\mathrm { x } = \mathrm { e } ^ { 2 }$ about the $\mathrm { x }$ -axis.

Evaluate the integral. - $\int _ { 0 } ^ { 1 / 4 } y \tan ^ { - 1 } 4 y \mathrm { dy } \quad$ (Give your answer in exact form.)

Integrate the function. - $\int \frac { \left( 49 - t ^ { 2 } \right) ^ { 3 / 2 } } { t ^ { 6 } } d t$

Evaluate the integral. - $\int \cos ^ { - 1 } x d x$

Integrate the function. - $\int \frac { d x } { \left( 25 x ^ { 2 } + 1 \right) ^ { 2 } }$

Use integration by parts to establish a reduction formula for the integral. - $\int ( \ln a x ) ^ { n } d x$

Solve the problem. -Find the length of the curve $y = \ln ( \sin x ) , \pi / 3 \leq x \leq \pi / 2$

Evaluate the integral. - $\int _ { 0 } ^ { \pi / 2 } \cos ^ { 2 } 3 x \sin ^ { 3 } 3 x d x$

Evaluate the integral. - $\int _ { 0 } ^ { \pi / 2 } \sin 4 t \sin 3 t d t$

Integrate the function. - $\int \frac { d x } { x \sqrt { 36 x ^ { 2 } - 49 } }$

Evaluate the integral. -Use the formula $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int f ( y ) d y , y = f ^ { - 1 } ( x )$ to evaluate the integral. $\int \cos ^ { - 1 } x d x$

Use integration by parts to establish a reduction formula for the integral. - $\int \sec ^ { n } x d x , n \neq 1$

Use various trigonometric identities to simplify the expression then integrate. - $\int \sin \theta \cos \theta \cos 5 \theta d \theta$

Find the area between $y = ( x - 3 ) e ^ { x }$ and the $x$ -axis from $x = 3$ to $x = 6$ .

Evaluate the integral. - $\int x ^ { 3 } \ln 4 x d x$

Evaluate the integral. - $\int y ^ { 3 } e ^ { - 3 y } d y$

Find the volume of the solid generated by revolving the region bounded by the curve $y = 2 \cos x$ and the $x$ -axis, $\frac { \pi } { 2 } \leq x \leq \frac { 3 \pi } { 2 }$ , about the $x$ -axis.

Find the volume of the solid generated by revolving the region in the first quadrant bounded by $y = e ^ { x }$ and the $x$ -axis, from $x = 0$ to $x = \ln 3$ , about the $y$ -axis.

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Exam 8: Techniques of Integration

Evaluate the integral by using a substitution prior to integration by parts. - ∫01xx+1dx\int _ { 0 } ^ { 1 } \frac { x } { \sqrt { x + 1 } } d x∫01​x+1​x​dx

Evaluate the integral. - ∫2xexdx\int 2 x e ^ { x } d x∫2xexdx

Find the volume of the solid generated by revolving the region bounded by the curve y=ln⁡xy = \ln xy=lnx , the xxx -axis, and the vertical line x=e2\mathrm { x } = \mathrm { e } ^ { 2 }x=e2 about the x\mathrm { x }x -axis.

Evaluate the integral. - ∫01/4ytan⁡−14ydy\int _ { 0 } ^ { 1 / 4 } y \tan ^ { - 1 } 4 y \mathrm { dy } \quad∫01/4​ytan−14ydy (Give your answer in exact form.)

Integrate the function. - ∫(49−t2)3/2t6dt\int \frac { \left( 49 - t ^ { 2 } \right) ^ { 3 / 2 } } { t ^ { 6 } } d t∫t6(49−t2)3/2​dt

Evaluate the integral. - ∫cos⁡−1xdx\int \cos ^ { - 1 } x d x∫cos−1xdx

Integrate the function. - ∫dx(25x2+1)2\int \frac { d x } { \left( 25 x ^ { 2 } + 1 \right) ^ { 2 } }∫(25x2+1)2dx​

Use integration by parts to establish a reduction formula for the integral. - ∫(ln⁡ax)ndx\int ( \ln a x ) ^ { n } d x∫(lnax)ndx

Solve the problem. -Find the length of the curve y=ln⁡(sin⁡x),π/3≤x≤π/2y = \ln ( \sin x ) , \pi / 3 \leq x \leq \pi / 2y=ln(sinx),π/3≤x≤π/2

Evaluate the integral. - ∫0π/2cos⁡23xsin⁡33xdx\int _ { 0 } ^ { \pi / 2 } \cos ^ { 2 } 3 x \sin ^ { 3 } 3 x d x∫0π/2​cos23xsin33xdx

Evaluate the integral. - ∫0π/2sin⁡4tsin⁡3tdt\int _ { 0 } ^ { \pi / 2 } \sin 4 t \sin 3 t d t∫0π/2​sin4tsin3tdt

Integrate the function. - ∫dxx36x2−49\int \frac { d x } { x \sqrt { 36 x ^ { 2 } - 49 } }∫x36x2−49​dx​

Use integration by parts to establish a reduction formula for the integral. - ∫sec⁡nxdx,n≠1\int \sec ^ { n } x d x , n \neq 1∫secnxdx,n=1

Use various trigonometric identities to simplify the expression then integrate. - ∫sin⁡θcos⁡θcos⁡5θdθ\int \sin \theta \cos \theta \cos 5 \theta d \theta∫sinθcosθcos5θdθ

Find the area between y=(x−3)exy = ( x - 3 ) e ^ { x }y=(x−3)ex and the xxx -axis from x=3x = 3x=3 to x=6x = 6x=6 .

Evaluate the integral. - ∫x3ln⁡4xdx\int x ^ { 3 } \ln 4 x d x∫x3ln4xdx

Evaluate the integral. - ∫y3e−3ydy\int y ^ { 3 } e ^ { - 3 y } d y∫y3e−3ydy

Find the volume of the solid generated by revolving the region bounded by the curve y=2cos⁡xy = 2 \cos xy=2cosx and the xxx -axis, π2≤x≤3π2\frac { \pi } { 2 } \leq x \leq \frac { 3 \pi } { 2 }2π​≤x≤23π​ , about the xxx -axis.

Find the volume of the solid generated by revolving the region in the first quadrant bounded by y=exy = e ^ { x }y=ex and the xxx -axis, from x=0x = 0x=0 to x=ln⁡3x = \ln 3x=ln3 , about the yyy -axis.